The fractional part of `2^(4n)/15` is `(n in N)` (a) `1/15` (b) `2/15` (c) `4/15` (d) none of these

Options

A. `1/15`
B. `2/15`
C. `4/15`
D. none of these

Explanation

To find the fractional part of \( \frac{2^{4n}}{15} \), we can follow these steps:

### Step 1: Rewrite \( 2^{4n} \)
We know that:
\[ 2^{4n} = (2^4)^n = 16^n \]
Thus, we can express our problem as:
\[ \frac{2^{4n}}{15} = \frac{16^n}{15} \]

**Hint:** Rewrite the expression using powers to simplify calculations.

### Step 2: Express \( 16^n \) in terms of \( 15 \)
We can express \( 16 \) as:
\[ 16 = 15 + 1 \]
So,
\[ 16^n = (15 + 1)^n \]

**Hint:** Use the binomial theorem to expand \( (15 + 1)^n \).

### Step 3: Apply the Binomial Theorem
Using the binomial theorem, we expand \( (15 + 1)^n \):
\[ (15 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} 15^k \cdot 1^{n-k} \]
This gives:
\[ (15 + 1)^n = \binom{n}{0} 15^0 + \binom{n}{1} 15^1 + \binom{n}{2} 15^2 + \ldots + \binom{n}{n} 15^n \]
Which simplifies to:
\[ 1 + n \cdot 15 + \frac{n(n-1)}{2} \cdot 15^2 + \ldots + 15^n \]

**Hint:** Identify the first term of the expansion as it will help in finding the fractional part.

### Step 4: Divide by 15
Now, we divide the entire expansion by 15:
\[ \frac{(15 + 1)^n}{15} = \frac{1}{15} + n + \frac{n(n-1)}{2} \cdot 15 + \ldots + \frac{15^{n-1}}{15} \]
The fractional part is determined by the first term:
\[ \frac{1}{15} \]
All other terms are integers since they involve \( 15 \) raised to some power.

**Hint:** Focus on the non-integer part after division to find the fractional part.

### Step 5: Conclusion
Thus, the fractional part of \( \frac{2^{4n}}{15} \) is:
\[ \frac{1}{15} \]

**Final Answer:** The correct option is (a) \( \frac{1}{15} \).